Embedding Algebraic Thinking throughout the Mathematics Curriculum
Author(s): G. PATRICK VENNEBUSH, ELIZABETH MARQUEZ and JOSEPH LARSEN
Source: Mathematics Teaching in the Middle School, Vol. 11, No. 2 (SEPTEMBER 2005), pp. 8693
Published by: National Council of Teachers of Mathematics
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G. PATRICK
VENNEBUSH,
ELIZABETH
MARQUEZ,
LOVE, ALGEBRA IS WHERE YOU FIND IT.
You can locateit almostanywherein the
middle school curriculumif you know
wheretolookandwhattolookfor.Butwith
so manydemandson ourtime,we oftenforget
to
look.We takeproblemsat facevalue,and we assumethata geometry
problemis justa geometry
orthata dataanalysisactivity
is onlyabout
problem
dataanalysis.Ifwe scratchbelowthesurface,
howforalgebraic
ever,we can findrichopportunities
in numberexplorations,
measurethinking
lurking
menttasks,andgeometry
investigations.
to
Today,manyschooldistricts
requirestudents
beginthe formalstudyof algebrain the eighth
a majorcomponentof the
grade. Consequently,
earlymiddleschoolyearsis dedicatedto fostering
algebraicthinking,
yetmiddleschoolstudentsare
PATRICKVENNEBUSH,patrick.vennebush®
verizon.net, formerlya developer at the
Educational TestingService (ETS), is the
Illuminations
projectmanagerat NCTM. He is
also theeditorofthe "Media Clips"columnin
theMathematicsTeacherjournal.ELIZABETH
is an assessment
MARQUEZ,
lmarquez@ets.org,
developer for the National Board for
ProfessionalTeachingStandardsat ETS. She
wasformerly
a teacherin theNorthBrunswick
whereshe received
SchoolDistrict,NewJersey,
AwardforExcellencein Math
thePresidential
and Science Teaching. JOSEPH LARSEN,
at
jplar@aol.com, is an educationalconsultant
ETS whohelpedtodevelopand nowassessesthe
a mathematics
middleschoolPraxisexams.He wasformerly
departmentchair at Council Rock School District in
Pennsylvania.This articleis theresultof theirworkwith
teachersacrossthecountry
in developing
TeacherAssistance
Packagesforalgebraand prealgebrainstruction.
86
and JOSEPH
LARSEN
alsoexpectedtolearnmanyconceptsandskillsthat
are not directly
relatedto algebra.Accordingto
Principlesand Standardsfor School Mathematics
shouldreceive
(NCTM2000),theAlgebraStandard
about one-third
of the emphasisin the middle
and
grades. (A graphicon page 30 of Principles
Standards"showsroughlyhow the ContentStandardsmightreceivedifferent
emphasesacrossthe
gradebands."This graphicis notmeantto dictate
the specificnumberof lessons for each topic
theyear,butitdoes suggestthattheAlthroughout
30
gebra Standardshouldreceiveapproximately
percentoftheemphasisinthemiddlegrades.)
Withso littletimereservedfortheAlgebraStandardintheearlymiddleschoolyears,howis itpossible to fosteralgebraicthinking
whilealso sufficientlyaddressingthe Standardsof Numberand
Measurement,and Data
Operations,Geometry,
andProbability?
Analysis
One effective
approachis to use richtasksthat
addressmorethanone Standard.Principlesand
do "not
Standards
statesthattheContent
Standards
curriculum
neatlyseparatetheschoolmathematics
intononintersecting
subsets.Becausemathematics
the areas
as a disciplineis highlyinterconnected,
describedby the Standardsoverlapand are integrated"(NCTM2000,p. 30).
Teachers oftenchoose rich activitiesforthe
oftopics
classroomthatallowforthe exploration
whenthey
frommorethanone standard.
However,
theconnections
betweentheStanfailto highlight
topromote
dards,teachersmaylose an opportunity
algebraicskillsin the middlegrades.This article
identifies
someofthealgebraimplicit
within
activitiesthatemphasizeotherconcepts,connectstraditionalalgebraproblems
totheotherfourStandards,
andoffers
formodifying
so that
activities
strategies
theywillbe abletofoster
algebraicthinking.
MATHEMATICS TEACHING IN THE MIDDLE SCHOOL
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WhatIs Algebraic
Thinking?
AND MANY
MENTIONTHE WORD "ALGEBRA"
- thinkof
teachers
included
students
and
people
variablesacx's and equationsand manipulating
a
of
rules.
to
set
manipAlthough
symbolic
cording
one ofthemostimportant
ulationis arguably
parts
ofalgebra,itis nottheonlypart.Algebraic
thinking
muchmore.
involves
Algebrais oftendescribedas thestudyofgenerInsteadof dealingsolelywith
alized arithmetic.
itfocuseson operations
numberand computation,
and processes.Greenesand Findellsuggestthat
involve
thebigideasofalgebraicthinking
represenof
balance,
tation,
meaning
reasoning,
proportional
and
and inductive
and functions,
variable,patterns
deductivereasoning(Greenesand Findell1998).
contendsthatalgebraicthinking
Kriegler
Similarly,
combinestwo veryimportant
components:
algeand funcbraicideas,including
variables,
patterns,
tools,specifically
tions,withmathematical
thinking
(e.g., diagrams,
problemsolving,representation
words,tables),and reasoning(Kriegler2004). In
that
is usedinanyactivity
short,algebraicthinking
of
the
with
one
combinesa mathematical
process
such
as
in
ideas
understanding
patalgebra,
big
with
situations
ternsand functions,
representing
models,and analyzsymbols,usingmathematical
foralgebraic
As
such,
opportunities
ing change.
as
illustrated
in
arise
contexts,
by
many
thinking
theexamplesthatfollow.
ina Number
Thinking
Algebraic
Sense Task
SYSTEMIMPLICNUMERATION
THEPLACE-VALUE
of
the
basic
some
conceptsofalgeitlyincorporates
number
of
and
the
bra,
rely
operations
algorithms
heavilyon the"lawsofalgebra"(NationalResearch
Council2001,p. 256).Itis notsurprising,
then,that
On each turn,a playerspinsthetwospinnersshownbelow.
The scorefortheturnis foundbytakingtheproductofthe
numberson whichthespinnersstop.Forexample,ifthefirst
spinnerstopson -5 and the secondspinnerstopson 2, the
scoreforthatturnwouldbe -5 x 2,or-10.
totalis keptbyaddingthescoreforeach turn,and
A running
theplayerwiththehighesttotalafter10turnsis thewinner.
Fig. 1 The rules forthe "positivelynegative" game
activities
dealingwithnumber
manymiddle-grades
even if
and operationsinvolvealgebraicthinking,
varithetaskdoes notspecifically
targetpatterns,
ables,orotheralgebraicideas.
thatcombinesnumber
One classroomactivity
is the"positively
senseandalgebraicthinking
negawithIntegers
tive"game,takenfromOperating
(ETS
2003c);therulesforthegamearegiveninfigure1.
involvesthemultiplication
This stimulating
activity
ofpositiveandnegative
andaddition
integers;
open
andthistopicusuan introductory
algebratextbook
fewpages.As students
play
allyappearson thefirst
unawarethattheyarepracthisgame,theyareoften
ticingbasic operationsand developingalgebraic
students'
skillsin
ideas.The tasksubtlyimproves
mathematics.
is an exampleofa questionthat
The following
to genIt asks students
fostersalgebraicthinking.
eratea sequenceofscoresforthe"positively
negaoutcome.
tive"gamethatwouldleadtoa particular
VOL. 11, NO. 2 • SEPTEMBER 2005 87
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After
Jacoband Simonehad each taken7 turns,
63to12.Simonecouldnotbelieve
Jacobwasleading
Showa sequence
thatshehadscoredonly12points!
ofscoresthatwouldyielda totalscoreof12points
from
ETS 2003c,p. 11)
7 turns.
after
(Adapted
becauseit
Thisquestioninvokesalgebraicthinking,
combinestheprocessofproblemsolvingwiththe
to
It also forcesstudents
algebraicidea ofpatterns.
thinkaboutissuesofbalance- one ofthebigideas
- byconsidering
addilistedbyGreenesandFindell
tiveinverses,and howpositivevaluescan be canvalues.
celedbynegative
When the "positivelynegative"spinnersare
as shownin
spun,thereareninepossibleproducts,
thegridbelow:
6
4-5
14-56
2
8
-10
12
-3
-12
15
-18
Task
and a Geometry
Thinking
Algebraic
THE DECREASINGVOLUME TASK,TAKEN FROM
Volumeand Dimension(ETS 2003b),iniExploring
whichdimension
tiallyasks studentsto determine
ofa5x7xl5 boxshouldbe decreasedby1 unitto
producethegreatestdecreaseinvolume.Students
are thenasked to generalizewhichdimensionof
box shouldbe reducedby1 unitto
anyrectangular
decreaseinvolume.
yieldthegreatest
For the specificcase of a 5 unitx 7 unitx 15
unit box, the typicalsolutionstrategyinvolves
studentsproceedas
testingeach case. Generally,
follows:
• Ifthe5-unit
sideis reduced,thenewboxwillbe
4x7x15, andhavea volumeof420cubicunits.
• Ifthe7-unit
sideis reduced,thenewboxwillbe
5x6x15, andhavea volumeof450cubicunits.
• Ifthe15-unit
sideis reduced,thenewboxwillbe
5 x 7 x 14,andhavea volumeof490cubicunits.
studentsconcludedthat
Fromtheseobservations,
desidewouldyieldthegreatest
the5-unit
reducing
creaseinvolume.
Studentsmightnoticethateach numberin the
bottomrowis equal to theoppositeofthesumof
in thefirst
thetwonumbersaboveit;forinstance,
column,-12 is the oppositeof4 + 8. Recognizing
this patternis usefulin answeringthe question
aboutJacoband Simone,becausethreeturnswith
Twosetsof
scoresof4,8,and-12 resultin0 points.
scoreof
thesethreescoresas wellas an additional
12pointswillyieldthedesiredresult:
4 + g + (-12) + 4 + 8 + (-12) + 12= 12
othersetsofthreescoreswillresultin
Similarly,
0 points:(-5, -10, 15), (6, 12,-18), (4, 6, -10), and
withan
(6,6,-12). Anytwoofthesesets,combined
willyielda totalscoreof
turnof12points,
additional
workatthebot12pointsafter7 turns.The student
tomoffigure2 uses thisideaofbalancetoobtaina
correctsequenceforSimone'sspins.
theidea
ofsymbolic
In thecontext
manipulation,
tothe
tocorrespond
ofbalanceis generally
thought
rule"Anything
youdo toone sideofthe
procedural
anditis imporequation,
youmustdo totheother,"
thisrulewhendetantforstudentsto understand
a broader
withsymbols.However,
velopingfluency
tounderstanding
viewofbalancerefers
equivalence
and opposites.Althoughthe "positively
negative"
with
to
students
proficiency
acquire
helps
game
it
enables
them
also
and
numbers,
negative
positive
ofbalance.
todevelopan intuitive
understanding
88
Fig. 2 A studentsolutionforthe Simone and Jacob problem
using the idea of balance
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Based on theexploration
ofa5x7xl5 box,the
Volume
task
Decreasing
mayappearto be solely
even
the
Indeed,
geometric.
generalcase regarding
a box ofanysize can be solvedwithgeometric
reawhoseworkis
soning,as was donebythestudent
showninfigure3. However,
thetaskextendstoalbecausestudents
areaskedtogengebraicthinking
eralizetheirfindings.
The student
workinfigure4
does notuse algebraicsymbols,
butitdoes use inductivereasoningto reacha conclusion.
Usingthe
previousresultsfromthe5 x 7 x 15 box,as wellas
boxeswithdimensions
of3 x 4 x 5 and
considering
5x2x8, the studentrecognizesa patternfrom
threespecificexamples.Moretothepoint,thestudent'sworkinfigure5 exhibitsobviousalgebraic
inthatthestudent
usedvariablesto reprethought
sentandanalyzethesituation.
The student
pictured
thethreeslicesthatcouldbe removed
fromthebox
andrepresented
theirvolumeswithformulas
using
theslice
/,wyand/г.If/гis theshortest
dimension,
withdimensions
w x I xl has thegreatest
volume.
thestudent
concludedthattheboxwith
Therefore,
volumeIwh- wll has the greatestdecreasefrom
theoriginal.
Thisis a generalized
formofthestrategyused infigure4. Sincealgebrais "thegeneralizationofarithmetic,"
thissolutionclearlydemonstratesa greatdealofalgebraicthinking.
The conclusion
infigure4 is based on boxesof
severalsizes,whereastheconclusion
infigure5 is
based on everypossiblebox size,Ixwxh. Consequently,even thougheach studentreceivesfull
creditforhis orherwork,theexplanation
givenby
thestudent
infigure5 is morecomplete.
Showing
students
an algebraicapproachto solvingproblems
like thismaybe used to introducethe powerof
In addition,
thisproblem
symbolicrepresentation.
canbe revisited
as students
acquirealgebraicskills
andas theyencounter
othersituations
aboutwhich
theyareaskedtogeneralize.
Fig. 3 A geometricsolutionto the Decreasing Volumetask
Fig. 4 An inductivereasoning solutionto the Decreasing Volumetask
inan AlgebraTask
Data Analysis
SOMETIMESTHE CONNECTION
BETWEENALGEbraandotherStandards
turnsup whereyouwould
least expectit.The Odd Integersproblembelow
was givento a groupofstudents
to determine
their
totranslate
intosymbolsand
wordproblems
ability
tosolveequations.
The sumoffourconsecutive
oddintegers
is 112.
What is the greatestof these fourintegers?
(MATHCOUNTS2003,p. 51)
Oneteacherwhousedthisproblem
expectedstudentstorepresent
eachofthefouroddnumbers
with
an algebraic
setupan equation,
solvefor
expression,
Fig. 5 An algebraic solutionto the Decreasing Volumetask
andinterpret
theresults.
a variable,
Figure6 shows
thatusedthisexpectedmethod,
onesolution
as well
as twoothersuccessful
solutions.
SolutionA exhibitsa strategy
typicalofwhatwe
and
mightexpectfroma first-year
algebrastudent,
solutionВ employsa guess-and-check
Alstrategy.
solution
В reliesentirely
on
thoughequallycorrect,
numbersense and computation.
SolutionC, aldemonstrates
numbersenseprincithoughatypical,
amountofalgebraicthinking.
plesanda significant
VOL. 11, NO. 2 ■ SEPTEMBER 2005 89
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Fig. 7 A representationshowingthe average of fourconsecutiveodd integers
Fig. 6 Solutions to the Odd Integersproblem
The firststep (dividing
by 4) findstheaverageof
In stating
thatthe"third
number
thefournumbers.
is 28 + 1,"thestudentseemsto use a variableiman unknown
theaverto represent
plicitly
quantity:
age ofthefournumbersis x,andthethirdnumber
is x + 1,usingthesymmetry
ofodd numberson eithersideofthemean.
The idea ofaveragingis oftendescribedas an
"eveningout"of data: take a littlefromthispile
to averages,soandadd ittothatpile.Byreferring
lutionС connectsalgebrato dataanalysis.Each of
theintegerscan be represented
byan expression,
90
an arrangement
ofalgebratiles,or a diagram.As
figure7 shows,thealgebraicaverageofthefour
integersis x + 3: removea squarefromx + 4 and
x+ 6
add ittox + 2, andremovethreesquaresfrom
andadd themtox. Numerically,
theaverageofthe
4 = 28.As a result,
x + 3 = 28,
fournumbersis 112-s=
whichmeansthatx 25, and the greatestofthe
fourintegersis x + 6, or31.
The Odd Integersproblemis a valuabletaskfor
middleschool students,since it invitessolutions
thatinvolvemultiple
strandsand enablesteachers
to highlight
how one problemcan be solvedwith
variousstrategies.
Unlikethe"positively
negative"
game,whichis a numbersense taskthatincorporates algebraicideas, and DecreasingVolume,
whichintegrates
algebraicconceptsintoa geometan
rictask,theOdd Integersproblemis foremost
- onethatmightappearina typical
algebraproblem
algebracourse- thatallowsfortheexeighth-grade
ofa topicfromdataanalysis.
ploration
Tasksto Promote
Modifying
Thinking
Algebraic
ANOTHER EXAMPLE OF THE CONNECTION BEtweendata analysisand algebra can be seen in the
Bookworms
taskfrom
Data and Making
Analyzing
Predictions
(ETS 2003a).As showninfigure8, the
taskpresentsdatacollectedfromstudentsurveys.
mostofthequestionsassess a student's
Although
topredataanalysisskills,question4 asks students
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dietresultsfortheentireschoolpopulation.
A solutionthatemploysalgebraicthinking
forquestion4
oftheBookworms
taskis shownin figure9. The
methodused infigure10, althoughnotstrictly
algebraic,also requiresthatstudentsapplyproportionalreasoning,whichis neededformanyalgebraictasks.
infigures9 and 10 are bothcorThe solutions
to showingstudents
rect,andthereare advantages
thevalueofeach method.On theone hand,bynot
a particular
solutionstrategy,
thistaskis
requiring
valuableforexpandingstudents'problem-solving
skills.Ontheotherhand,theproblem
canbe usedto
assess a student's
totranslate
wordproblems
ability
to equationsand to manipulate
To obtain
symbols.
evidenceforthispurpose,
itmaybe necessary
toask
a follow-up
suchas thefollowing:
question,
Explainhowtheproblemmightbe solvedusing
a proportion.
This minormodification
makestheBookworms
taskappropriate
forassessingproportional
reasonHowing,whichis essentialforalgebraicthinking.
ever,substantial
changesare occasionally
required
to makea tasksuitableforstudents.
For example,
the finalpartof the DecreasingVolumetask reto explainthesolutionforanysize
quiresstudents
forstudents
box,butsuch a taskmaybe difficult
who have minimalexperiencein generalizing
results.Addingthe information
shownin table 1
shouldhelp studentsidentify
a patternas dimensionsare changed.Usingtheresultsfromtable 1,
the studentcan be askedto explainwhichdimension shouldbe reducedby 1 unitto producethe
decreaseinvolumeforanysizebox.
greatest
As shownbythelastrowofthetable,themodificationcanincludevariables,
forthediscovallowing
and assessmentof symbolic
ery,reinforcement,
andmanipulation
skills.
representation
as showninfigure6, theOdd Integers
Similarly,
canbe solvedusingalgebra,
number
problem
sense,
ordataanalysis
To ensurethatthetaskretechniques.
toapplyalgebraic
quiresstudents
thinking,
anyofthe
additions
ormodifications
couldbe used:
following
• Ifn is an oddinteger,
whatexpressions
couldbe
usedtorepresent
thenextthreeoddnumbers?
• Fourconsecutive
odd integersare represented
byn,n + 2, n + 4,andn + 6,andthesumofthese
fournumbersis 112.Solvetheequationn + in +
theval2) + (n + 4) + (n + 6) = 112to determine
ues ofthefourintegers.
• The smallestoffourconsecutive
odd integers
is
n,andthesumofthesefournumbersis 112.In
the threeblanksbelow,writeexpressionsthat
дО
jÖ'
'j-+^t
^^^
f*^^^^^]
'^ J
^^^^
'^^^^^
BOOKWORMS
i
tooka surveyoftheirclassmatesto determine
Juananc*Britney
whattypeofbooks theymostenjoyreading.Juansurveyedthe
boys,and Britney
surveyedthegirls.The resultsare shownin
thetablebelow.
TypesofBooksThatStudentsMostLiketo Read
Type of Book
Action
Juan (Boys)
8
Britney(Girls)
18
Mystery
Science Fiction
8
13
18
18
Nonfiction
6
11
Total
Total
Q
1. Completethetableabove.
2. On a separatesheetofpaper,makean appropriategraphicalrepresentation
ofthe
data collectedbyJuan.
3. What percentof thegirlssurveyedbyBritneylikeactionbooks best? Show how
you foundyouranswer.
I
(Л
cr
Purchase answer to see full
attachment